6x(x+3)+x=7x+2+1

Solution for 6x(x+3)+x=7x+2+1 equation:

Simplifying
6x(x + 3) + x = 7x + 2 + 1

Reorder the terms:
6x(3 + x) + x = 7x + 2 + 1
(3 * 6x + x * 6x) + x = 7x + 2 + 1
(18x + 6x2) + x = 7x + 2 + 1

Reorder the terms:
18x + x + 6x2 = 7x + 2 + 1

Combine like terms: 18x + x = 19x
19x + 6x2 = 7x + 2 + 1

Reorder the terms:
19x + 6x2 = 2 + 1 + 7x

Combine like terms: 2 + 1 = 3
19x + 6x2 = 3 + 7x

Solving
19x + 6x2 = 3 + 7x

Solving for variable 'x'.

Reorder the terms:
-3 + 19x + -7x + 6x2 = 3 + 7x + -3 + -7x

Combine like terms: 19x + -7x = 12x
-3 + 12x + 6x2 = 3 + 7x + -3 + -7x

Reorder the terms:
-3 + 12x + 6x2 = 3 + -3 + 7x + -7x

Combine like terms: 3 + -3 = 0
-3 + 12x + 6x2 = 0 + 7x + -7x
-3 + 12x + 6x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
-3 + 12x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-1 + 4x + 2x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-1 + 4x + 2x2)' equal to zero and attempt to solve:

Simplifying
-1 + 4x + 2x2 = 0

Solving
-1 + 4x + 2x2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-0.5 + 2x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 2x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 2x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 2x + x2 = 0 + 0.5
2x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
2x + x2 = 0.5

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 0.5 + 1

Reorder the terms:
1 + 2x + x2 = 0.5 + 1

Combine like terms: 0.5 + 1 = 1.5
1 + 2x + x2 = 1.5

Factor a perfect square on the left side:
(x + 1)(x + 1) = 1.5

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(x + 1) equal to 1.224744871 and -1.224744871.

Subproblem 1
x + 1 = 1.224744871

Simplifying
x + 1 = 1.224744871

Reorder the terms:
1 + x = 1.224744871

Solving
1 + x = 1.224744871

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + x = 1.224744871 + -1

Combine like terms: 1 + -1 = 0
0 + x = 1.224744871 + -1
x = 1.224744871 + -1

Combine like terms: 1.224744871 + -1 = 0.224744871
x = 0.224744871

Simplifying
x = 0.224744871

Subproblem 2
x + 1 = -1.224744871

Simplifying
x + 1 = -1.224744871

Reorder the terms:
1 + x = -1.224744871

Solving
1 + x = -1.224744871

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + x = -1.224744871 + -1

Combine like terms: 1 + -1 = 0
0 + x = -1.224744871 + -1
x = -1.224744871 + -1

Combine like terms: -1.224744871 + -1 = -2.224744871
x = -2.224744871

Simplifying
x = -2.224744871

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.224744871, -2.224744871}
Solution
x = {0.224744871, -2.224744871}